How We Get Our Students to Read the Text Before Class
(MAAOL Version)

Abstract: We describe an email-based approach to reading assignments
that has been very effective in getting our students to read the text
before class. The dramatic impact this approach has had on
our courses is explained through sample assignments and student
responses. We also share the results of seven semesters of student
evaluations and address some implications of using these assignments.

1. Introduction

When students read the text before class, the fundamental nature of
class meetings is changed. The students arrive familiar with basic
concepts and definitions, providing more class time to address the
major ideas and subtleties of the mathematics. In addition, the
instructor is no longer viewed as the sole source of content for the
course, and this encourages greater independence, and more lively
interactions, among students. The challenge, of course, is getting
students to consistently read the text before class for the entire
semester.

Unfortunately, few of our students have experience reading a math text, and
most treat the book as a reference to use after the professor has
presented new material. To counter these habits, one approach is to simply
give a reading assignment for each class meeting. In our experience, most
students are unlikely to read consistently for the entire semester unless
there is some form of direct evaluation to keep them accountable. Since
any assessment during class interferes with the main goal of freeing class
time to discuss mathematics, it is important that such a method use
alternate means to promote the activity of reading. In this article, we
describe how email-based reading assignments have transformed a broad range
of our courses, including Introductory Statistics, Single and Multivariable
Calculus, Linear Algebra, and Geometry.

2. Our Goals, both Big and Small

One of the challenges to learning mathematics is that understanding is
often built in stages, and one's perspective deepens upon revisiting
concepts a second, third, nth time. If class time may be spent
on students' second exposure to basic terminology and elementary
examples, then the class is able to get to deeper mathematics more quickly
and in more detail. Indeed, this moves a class session from simply
introductory lectures to a time when elementary ideas are clarified (as
necessary) and expanded upon.

In addition, we strive in our courses to promote students' logical
reasoning and writing skills. It is often a shock to first year
mathematics students that the instructor would expect them to write (and in
complete sentences!) about mathematical ideas. While one can encourage
such activity on homework and exams, it is ideal to have as many different
activities as possible in which to develop writing skills. By reading a
mathematics textbook for content, as well as through responding to
questions about the reading, we aim to raise the level of students'
writing, along with improving their reading skills.

While these goals are broad and perhaps ambitious, our desires for
individuals on a day-to-day basis are quite modest. We want the students to
be familiar with past and upcoming terminology and to have a rough idea of
the basic concepts from each section. If each student spends some
time reading and preparing for class, then we believe that many of the
bigger goals will be accomplished. Finally, we also desire to reward our
students for their effort, while making sure that the approach to reading
is perceived as reasonable by both student and instructor.

3. The Details of the Assignments

We place the reading assignments on a course webpage, usually in month or
week long segments. This frees class time from announcing or distributing
the assignments and makes the assignments conveniently available to
students outside of class. The posting lists the specific section(s) to
read, which parts should be emphasized, and which can be skipped, if
any. There are also several basic questions that the student should be able
to answer after completing the reading. The questions serve to focus the
students' reading and give them feedback on their level of comprehension;
students email their responses to the instructor before the following class
meeting. This gives the instructor feedback on the level of the students'
understanding before class and allows the instructor to make
adjustments as necessary.

As an example, the following is an assignment
from Calculus II; the course text was [1].

For February 17

Section 3.8 Inverse Trigonometric Functions and Their Derivatives

To read: All, but you can skip the section on
Inverse Trigonometric Functions and the Unit Circle

Reading Questions:

What is the domain of the function arccos(x)? Why?

Why are we studying the inverse trig functions now?

Find one antiderivative of 1/(1+x^2).

We have found that a binary grading scheme works well for the assignments:
a student earns a 1 for sincerely attempting to answer the questions
(independent of whether the answers are correct), or receives a 0 if no
such attempt is made. In addition, the assignments count for 5% of a
student's final grade in the course. This assessment method has several
advantages. First, it emphasizes that a major point of the assignments is
making an honest effort, and also reduces the stress that many students
feel toward assignments in general. Further, this scheme makes the
assessment of the assignments fairly easy for the instructor. For a class
with 30 students, it takes approximately 20 minutes to read and record a
given day's responses from the class. Another effective tactic has been to
require the students to enter a specific subject line in their email
messages. The instructor can then use an email filter to move messages
with that subject line into a specific folder and generate an automatic
response, letting the student know that the assignment has been received.

The student responses are always informative, and they often provide an
excellent starting point for class discussion. We choose several of the
best responses to each assignment and place them on a temporary webpage.
By displaying these responses at the beginning of class, students can
compare their own thoughts on the reading, as well as see the work of some
peers. This activity sparks both questions and responses, often resulting
in discussion of key subtleties in the material. By archiving these web
pages, students are also able to view the responses after class at any
point later in the term.

4. Sample Reading Questions and Student Responses

4.1 From Calculus II

In our calculus sequence, we do not cover inverse trigonometric functions
until Calculus II. The sample assignment in
Section 3 came after we had discussed numeric integration but before we had
covered any techniques of antidifferentiation.
The student responses that were displayed during class were:

The domain of the arccos(x) is [-1,1], because the range of the
cos (its inverse), is [-1,1]. A.V., First-Year

We are studying inverse trig. functions now because by knowing
the derivatives of these functions, we will be able to calculate more
definite integrals using the FTC (Fundamental Theorem of Calculus).
A.C., Sophomore

One antiderivative of 1/(1+x^2) is arctan(x) + 3.
M.K., First-Year

These answers all show that the students understand the fundamental issues
raised by the questions. A.V.'s response shows an understanding of the
relationship between the range of a function and the domain of its
inverse. A.C. gives a nice justification for why we are introducing
the inverse trigonometric functions at this point in the course, and
M.K. demonstrates the important point that the antiderivative is not
unique.

Obviously, not all students gave such precise answers to all questions. In
fact, M.K. completely missed the motivation for studying the inverse
trigonometric functions. However, most students' misunderstandings were
minor and were cleared up at the beginning of the class. This allowed
enough time in a 50 minute class to derive the derivatives of arcsin(x) and
arctan(x) and to give the students 15 minutes of in-class work. Without
knowing the students' level of understanding before class, it is highly
unlikely that we could have accomplished as much in one class meeting. With
no assessed reading assignment, more time would have been spent on
introductory material and motivation. Assessing the reading in class would
not only eat into class time but would also make it more difficult to
adjust the class meeting based on the students' responses.

4.2 From Geometry

The following assignment from early in the semester centered on the
introductory section to the study of Euclidean motions of the plane. While
the material had a new geometric perspective to students, they should have
been familiar with many of the basic ideas from prior courses. The course
text was [3].

For Monday, January 24

Reading Assignment: Section 2.1 (all)

Reading Questions:

What is the difference between a mapping and a function?

Is every mapping a transformation? (Explain, including a
description of a transformation.)

Does every transformation have an inverse? Why or why not?

The following were among the student responses shared in class:

Mapping means that every element a of A has a unique element
b of B that is paired with a. A function is a set of ordered pairs
(a,b) with no two different pairs having the same first element.
Therefore, they have similar definitions. The main difference is
that Mapping is the term used in geometry, rather than the term
Function. M.M., Junior

No every mapping is not a transformation. A transformation
is when the (x,y) are altered or reversed in some way. It consists
of one-to-one and onto functions. When you reverse the pairs, it
does not always result in a mapping. Other than the reversing of
pairs, a mapping is a transformation. S.S., Junior

Every transformation has a unique inverse. Since a
transformation is one-to-one and onto, it means that there is
exactly one element in A that that matches with one element in B.
So no matter if you are going to B from A or to A from B, there
will always be a corresponding element in the second set. [It's
kinda like "for every action, there is an equal and opposite
reaction.''] L.S., Junior

M.M. shows here that he has good command of the basic ideas in question 1;
not only are the definitions "similar,'' but in fact they are identical.
This was the point of the question. Similarly, in question 3,
L.S. demonstrates an understanding of the fact that all transformations are
invertible. Her response includes a nice description of a one-to-one
correspondence that students in class found a good explanation.

In question 2, however, S.S. reveals a less than complete understanding of
the definition of a transformation. Such a response offers many
opportunities in class: is there a difference in saying "every mapping is
not a transformation'' and "not every mapping is a transformation''? The
response includes some of the main ideas involving one-to-one and onto
functions; the lesson is that sometimes an imperfect response can provide
an excellent learning moment for the entire class, particularly if several
students shared in the difficulty. All three responses enabled us to have
a brief, but important, discussion of how important precise language is in
mathematics.

In reviewing the reading responses to these three questions, it was clear
before class that most students had a solid grasp of the material. A few
short minutes at the start of class were used to make certain the
terminology was clear to all, and from there we were able to quickly
develop more in-depth ideas related to the geometric concepts we were
studying with the Euclidean motions. Had class instead begun with the
question "What is the definition of a function?'', followed by introducing
the term "mapping'', and then "transformation,'' it is certain that a much
more lengthy segment of time would have been devoted to elementary review.

4.3 General Remarks on Student Responses

We have observed several unexpected trends while reading our students'
responses. First, students tend to be more verbose via email than they are
in handwritten exercises. Certainly a part of this is the ease of editing
and expanding their responses at the keyboard. Secondly, the lack of
mathematical symbols in email is actually a large advantage since it forces
the students to explain their thought process in prose. Finally, providing
another regular mechanism for communication gives students who are
typically quiet in class an outlet to express their insights and share them
with the rest of the class when their email is displayed at the beginning
of class.

5. Data from Student Responses to Supplementary Evaluations

In each class where this approach to reading assignments has been used, we
have conducted a supplementary anonymous evaluation to gain further student
feedback. The students were given four options

(1) Strongly disagree
(2) Disagree
(3) Agree
(4) Strongly agree

to respond to the statements:

The reading assignments were helpful in understanding the
course material.

The reading assignments were useful in preparation for
the class meetings.

The reading questions were helpful in
focussing my reading.

I would have regularly read the text before class without
the reading assignments.

Table 1. Mean Responses to Supplementary Evaluations

Term

Course

Q1Understanding

Q2Preparation

Q3 Focussing

Q4
Read without

Spring 97

Calculus I

2.9

3.0

3.0

n/a

Fall 97

Calculus I

3.2

3.3

3.2

n/a

Calculus II

2.8

3.0

3.2

n/a

Multivariable

2.8

3.3

3.2

n/a

Spring 98

Calculus II

3.2

3.2

3.5

1.9

Fall 98

Calculus I

3.1

3.2

3.1

2.3

Linear

3.2

3.2

3.4

1.7

Multivariable

3.3

3.4

3.4

2.1

Spring 99

Calculus II

3.1

3.3

3.4

2.1

Fall 99

Calculus II

3.0

3.1

3.2

2.0

Linear

3.1

3.3

3.1

2.0

Spring 00

Intro Stats I

3.2

3.2

3.2

2.1

Intro Stats II

2.9

3.1

3.1

2.3

Geometry

3.4

3.5

3.5

1.9

Table 1 demonstrates that on average, students agree with the statements
that the reading assignments were helpful in understanding course material,
even moreso in preparing for class meetings, and likewise in helping them
focus their reading. In addition, students generally disagreed with the
statement "I would have regularly read the text without the assignments.''
This data supports what has been our consistent experience with this
approach.

Not only did students believe that the reading assignments were a good
idea, they actually did the reading! The first column of Table 2 shows the
students' response to the question:

On average, how much time did you spend on each reading assignment?
(1) 0--15 mins
(2) 15--30 mins
(3) 30--45 mins
(4) 45--60 mins
(5) More than an hour

The latter two columns of Table 2 show the mean percent of respondents per
assignment and the median percent of assignments completed per student.
(We use the median to reduce the influence of the small number of outliers
who completed few of the assignments.)

Table 2. Time per Assignment
and Response Rates

Term

Course

Mean Time/ Student

Mean Response/
Assignment (%)

Median Completed/ Student (%)

Spring 97

Calculus I

2.5

82

86

Fall 97

Calculus I

1.9

74

88

Calculus II

1.8

78

88

Multivariable

2.2

73

70

Spring 98

Calculus II

2.0

82

88

Fall 98

Calculus I

2.0

80

89

Linear Alg

2.0

84

90

Multivariable

1.9

83

96

Spring 99

Calculus II

2.2

83

92

Fall 99

Calculus II

1.9

72

86

Linear Alg

2.0

75

86

Spring 00

Intro Stats I

2.6

82

83

Intro Stats II

2.8

82

92

Geometry

2.7

89

96

Overall, we observe that on average students spent about 30 minutes on a
given reading assignment. In addition, consistently at least 75% of each
class completed and responded to a particular set of questions. Moreover,
the final column indicates that for most students, the vast majority of the
overall collection of reading assignments was completed. These data,
together with the student comments regarding their opinion that the
exercises were effective, demonstrate the high level of student involvement
in this activity, and make plausible our claims that the efficiency of
class time was significantly improved. While we would prefer that every
student complete every reading assignment, we consider the approach very
successful when 80% of the students in an Introductory Statistics course
spend, on average, more than 30 minutes reading the text before the
material is discussed in class.

Finally, it is again students' own words that offer so much evidence
of their satisfaction regarding these assignments:

"I firmly believe I would not have read as thoroughly and would not
have been as prepared for class were it not for the reading
questions. They weren't a big deal to complete at all, and I feel
they were vital in my understanding of the course.'' --
Geometry

"I felt they were very helpful considering I tend to struggle with
math
courses. A very good idea!!'' -- Statistics

"Good stuff, helps to at least get a feel for the material before it
is covered, allows a slightly faster pace.'' -- Linear Algebra

"I felt the reading questions made me concentrate more on what I was
reading and (I) got more out of the reading than I otherwise would
have.''
-- Calculus II

"They were quite helpful. But it was sometimes frustrating if I
didn't
understand the material to have to wait until class to finally see
how
to do it.'' -- Calculus II

The last quote demonstrates what we are striving for: students who
are thinking about mathematics, working on mathematics, and cannot
wait to get to class.

6. Other Issues/Potential Pitfalls

There are some start-up costs to be aware of when using these
assignments. Writing the assignments can be a time-consuming affair, and we
have found that it is easiest to write several weeks, or a month, of
assignments at a time. This has required us to have our courses fairly
well-organized to assign specific readings this far in advance. One
advantage is that this has helped us keep a brisk pace in our courses and
keep up with our initial syllabus.

Text selection is extremely important when using these assignments since
the students will be reading the text as their first introduction to the
course material. The students' perception of the readability of the text,
as well as the choice of questions, can significantly affect their opinion
of the efficacy of the assignments. If the questions are simplistic, then
the students view the assignments as busy work; if the questions are too
difficult, then they add to the frustration that many students feel when
reading mathematics. Quite often, several semesters of minor adjustments
are required to fine-tune the questions.

We also feel that it is important to recognize that these reading
assignments add to the students' workload in the course. Since the
assignments keep the students engaged with the course material on a nearly
daily basis, they can serve a similar role to lengthy homework assignments.
It is important that these reading tasks not simply be added to the list of
things required of students, but that their addition is reasonably
accomodated in an overall vision for expectations of students.

There are, of course, problems that can arise when an assignment is
technology dependent, such as access to email, network outages, and student
apprehension about using the technology. Since network problems will
inevitably occur, we have told students that they can turn in their
assignments on paper before class if they have trouble accessing email the
night before the assignment is due. A bit of flexibility on the part of
the instructor seems sufficient to handle these minor challenges.

7. Conclusion

We find the overall atmosphere in our classes exciting with this approach.
Students read to learn mathematics . . . They explain their mathematical
ideas in prose . . . Discussions become more lively . . . The instructor
gets individual feedback on each student's understanding of concepts . . .
Class time is spent more efficiently . . . Deeper mathematics is considered
. . . Students even profess to like the assignments.

It sounds like everyone is winning! The approach has changed the
fundamental way we direct our students in learning mathematics, and does so
in a way with many important benefits. For all these reasons, we hope that
other instructors will join us in the endeavor. The reader is encouraged
to take a look at how an entire semester develops in this approach by
visiting our courses on the World Wide Web at